#198786

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60782	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1111 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(18,22,25,24)(19,26,28,27);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,18,20,24,21,25,29,22)(19,23,27,31,28,32,26,30);(1,4)(2,7)(3,10)(5,15)(6,14)(8,12)(9,13)(11,16)(17,19)(18,23)(20,27)(21,28)(22,30)(24,31)(25,32)(26,29);(17,20,21,29)(18,24,25,22)(19,27,28,26)(23,31,32,30)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105982);